# Can finitely many hermitian (positive-semidefinite) operators always be embedded into a small dimensional space preserving inner product?

Given $n$ hermitian (positive-semidefinite) operators $Q_1,\cdots,Q_n$ in finite dimensional Hilbert space (the dimension can be very large), is there a mapping $\phi$ maps $Q_i$ to $P_i$, which preserves inner product, i.e. $\langle P_i, P_j\rangle =\langle Q_i,Q_j\rangle$, and all $P_i$'s are hermitian (positive-semidefinite) operators staying in a smaller dimensional space, say $poly(n)$ ?

Further question, given $n^2$ real numbers $c_{ij}$ $1\leq i,j\leq n$, how to decide if there exist $n$ hermitian (positive-semidefinite) operators $P_1,\cdots,P_n$ satisfying $\langle P_i,P_j\rangle=c_{ij}$? If exists, what is the minimum dimension of operators?

For vectors, the questions above are trivial. I wonder if there are any known results for operators? Thanks.

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Since you are making a distinction between vectors and operators, it is not clear to me which inner product you are considering. –  Martin Argerami Jun 21 '11 at 15:06
Sorry, the inner product for operators are defined to be the trace of the product. –  Penghui Yao Jun 22 '11 at 2:05
it might be more interesting to study the case where the inner products are only approximately preserved.... –  Suvrit Jun 23 '11 at 4:02

Let $G$ be the Gram matrix, with $ij$-entry $\langle Q_i,Q_j\rangle$. Then the rank of $G$ is the dimension of the space spanned by the operators $Q_i$. So the answer to your first question is no. (The point is that this all works in any inner product space, changing from "vectors" to "operators" makes no difference.)
I just realize the question is trivial for vectors and Hermitian case because $n$ vectors or Hermitian matrices can always be put into $n$-dimensional space preserving inner product. So the more interesting case for me is positive-semidefinite matrices, it is just a convex cone, not vector space. Are the questions non-trivial in this case? –  Penghui Yao Jun 22 '11 at 2:21
This may be far from what you are interested in, but I should mention it: things become non-trivial if you consider operator spaces or operator systems: these are (possibly finite-dimensional) subspaces $W$ of bounded operators on a Hilbert space $B(H)$ together with the norms on all spaces of $n\times n$ matrices over $W$. There are several books on the subject, including books by G. Pisier, E. Effros and J.-J. Ruan as well as V. Paulsen. –  Dima Shlyakhtenko Aug 30 '11 at 17:33